Normal two-dimensional elliptic singularities

Abstract

Abstract. Given a weighted dual graph such that the canonical cycle K' exists, is there a singularity corresponding to the given weighted dual graph and which has Gorenstein structure? This is one of the important problems in normal surface singularities. In this paper, we give a necessary and sufficient condition for the existence of Gorenstein structures for weakly elliptic singularities. A necessary and sufficient condition for the existence of maximally elliptic structure is also given. Hence, the above question is answered affirmatively for a special kind of singularities. We also develop a theory for those elliptic Gorenstein singularities with geometric genus equal to three.

Numerical Gorenstein elliptic singularities

“…The elliptic singularities were introduced by P. Wagreich in [24]; (X, p) is said to be elliptic if χ(Z) = 0. The theory of those singularities were developed by Wagreich [24], H. Laufer [8], S.S.-T. Yau [29], [30], [31], [32], M. Tomari [20], [21] and others. Laufer proved that Gorenstein singularities with p g = 1 are topologically characterized, and that for those singularities, mult = max{−Z 2 , 2} and emb dim = max{−Z 2 , 3}.Yau introduced the elliptic sequence, which is a topological invariant of an elliptic singularity, and obtained many results similar to those above, under certain conditions.…”

Filtered Rings, Filtered Blowing-Ups and Normal Two-Dimensional Singularities with “Star-Shaped” Resolution

“…The criterions in [48] of Gorenstein property for elliptic singularities are in some sense analogous to ours.…”

On numerically Gorenstein quasi-simple elliptic singularities with 𝐶*-action

“…Also, for numerically Gorenstein elliptic singularities, he gave the definition of maximally elliptic singularities. Further, in [7], he found the condition for {X, x) to be maximally elliptic and gave the following definition. Definition 1.1 [7,Definition 3.1].…”

“…Further, in [7], he found the condition for {X, x) to be maximally elliptic and gave the following definition. Definition 1.1 [7,Definition 3.1]. Let it be the minimal good resolution.…”

“…graph is given as in (i). Then the embdim(Z, x) (= embedding dimension of (*,x)) = max(3, -Z2), where -Z2 = Y,"=l(b'~ 1) • Yau [7,Theorem 2.4] had obtained a fine condition for numerically Gorenstein elliptic singularities to be maximally elliptic. We can prove (ii-b) directly from his result.…”